Magnetic force

Two forms, both printed in the Magnetism group of the AP Physics 2 sheet:

They are one law, not two. A current is charge in motion, so the wire version is the charge version summed over the carriers in the length $\ell$ inside the field. In the first, $\theta$ is the angle between the velocity and the field; in the second, between the current direction and the field. Either way the angle is never measured from the surface or from the wire.

Direction comes from the [right-hand rule](/glossary/right-hand-rule), and it reverses for a negative charge. Check any answer against the definition: the force is perpendicular to both inputs, so a force lying in the plane of $\vec{v}$ and $\vec{B}$ is wrong before you check its sign.

It never does work. Work is $Fd\cos\theta$ with the angle taken between force and displacement. The displacement is along the velocity and the force is perpendicular to the velocity, so the cosine is zero at every instant. No work means no change in kinetic energy, so a magnetic field can turn a charged particle through any angle and cannot change its speed. It steers.

The CED fences the arithmetic for a moving charge to angles of 0, 90 and 180 degrees, permitting qualitative analysis of the rest. No such limit is written for the wire. Topic 12.2 carries the procedure.

A note on the name. The full Lorentz force is the electric and magnetic forces together, $\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$; what AP Physics 2 calls $F_B$ is its magnetic term alone. In a region with both fields the two act independently, and only the electric one can change a charge's speed.